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A formula for pi involving nested radicals

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Abstract

We present a new formula for pi involving nested radicals with rapid convergence. This formula is based on the arctangent function identity with argument \(x=\sqrt{2-{{a}_{k-1}}}/{{a}_{k}}\), where

$$\begin{aligned} {{a}_{k}}=\underbrace{\sqrt{2+\sqrt{2+\sqrt{2+\cdots +\sqrt{2}}}}}_{k\,\,\text {square}\,\,\text {roots}} \end{aligned}$$

is a nested radical consisting of k square roots. The computational test we performed reveals that the proposed formula for pi provides a significant improvement in accuracy as the integer k increases.

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References

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Acknowledgements

This work is supported by National Research Council Canada, Thoth Technology Inc. and York University. The authors thank the reviewers for constructive comments and recommendations.

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Authors and Affiliations

  1. Dept. Earth and Space Science and Engineering, York University, Toronto, M3J 1P3, Canada

    S. M. Abrarov & B. M. Quine

  2. Dept. Physics and Astronomy, York University, Toronto, M3J 1P3, Canada

    B. M. Quine

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  1. S. M. Abrarov
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  2. B. M. Quine
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Correspondence to S. M. Abrarov.

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Abrarov, S.M., Quine, B.M. A formula for pi involving nested radicals. Ramanujan J 46, 657–665 (2018). https://doi.org/10.1007/s11139-018-9996-8

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  • DOI: https://doi.org/10.1007/s11139-018-9996-8

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